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9.6 Notes AP

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7.7 solving right triangles okay we're still going to use all the trig stuff that we learned specifically for the first half of this lesson from before the test and actually everything that we learned before the test this whole chapter still applies and is put together into this lesson now sohcahtoa still applies sign is opposite over her please cosine is adjacent over hypotenuse in tangents opposite over adjacent we're going to use this in all of this know what we use this for was to find one side of a right mangle if we knew at least one angle and a side we can find the other two so we use this to find site or we're going to do today is we're going to use what's called in trig ratios and we're going to use this to find the angle measures of a right triangle so if we have a 3 4 5 triangle every 3 4 5 triangle has the same two angles we need to find out what those two angles are and that's what we're going to use here to find that so let a be an acute angle okay we're in angle a we're going to find that so angle a is acute of course because that's not a nice way and be have to be a cute the first one we're going to learn is inverse sine okay self sign is opposite over hypotenuse so the three over the 5 now inverse sine is still going to be the same thing sit over hypotenuse but we're going to use the inverse of sine to find the angle so what does that mean right here we have if sign a why so sine of angle a equals y that's going to be our 3/5 right there that's what that y stands for now we want to find angle so we want to get this a by itself so think about this algebraically if I have the sine of angle a equals why I want to get a bite self what do I need to stop I need to get rid of the sign whatever I do to one side I have to do to the other those cancelled so now I have a / y / son he tried to type in a number divided by sine in your calculator to make any sense cuz you got to take the sign up something well anytime that you have something it has an exponent one and you should have learned in Algebra 1 that if you move it across the fraction bar it changes from positive to negative or negative to positive it's positive so if I move it to the top it becomes negative so if I move the sign one on top it becomes sign negative one and then I still have my wife so it's no longer on the bottom it's on top and that is where we get this right here so this is the short and formula the measure of angle a equals sine of -1 that is called the inverse sine of the opposite over the hypotenuse now let's actually use that and find angle a here so the picture of angle a equals the inverse sine of the opposite which is 3 over the hypotenuse which is 5 now these buttons are on your calculator they are above the sun cosine and tangent if you look you'll see son -1 cosine and tangent -1 so hit sign in sign negative one comes up with a princey and then type in 3/5 closure princy hit inner you get 36.8 69 so far into the 10th that is going to be 36.9 degrees so this has to be 36.9 degrees now if I'm bi track that from 9 okay so that's inverse sine inverse cosine same thing if you look cosine is why so if I move the cosine over here shortcut move the sign over divide it becomes the inverse so I have the measure of angle a is the inverse the sign of cosine is adjacent over hypotenuse so if I do that here the measure of angle a is the cosine negative one Jason in this one to A is for the hypotenuse is still 5 so for over five type that in your calculator hit second cosine type in 4/5 closure prints he hit inner you should get the same thing cuz we're solving for the same angle just using a different way 36 point degrees okay it's going to be the same thing for tangent tangent is opposite 3 or adjacent for so let's go down and that one if tan a equals y then move the tan over here it becomes inverse tan of why so we get this formula so the measure of angle a equals the inverse tan opposite was three adjacent was for type that in your calculator II 3 / foreclosure princi hit in her and you still get the same measurement for mangle a cuz angle a has not changed 36 play greased that's how you use inverse now let's try a few alright soap find each angle measure to the nearest so right now we have 10 of a equals .404 okay now what we need to do is we need to get to say by itself so we move the tangent over here becomes the inverse tangent so a equals 10 of -1 of 4040 can you when you move the tan across the equal sign it becomes the inverse type that in your calculator exactly like it looks and you should get other says to the nearest degree so we're going to round up 21.99 to 22° okay after getting something weird and you know you're typing it in right make sure you're in degrees and not Radiance number to we need to move the cosine over here so be equals inverse cosine of .46 9 V type that in your calculator exactly like it looks and you should get 60 1.99 so free Roundup you're going to get 62° now these next two I want you to pause the video I want you to work him out and then start it again and see if you got them right okay so you should have for number 3 gotten you equals 56 degrees and 4 number for a equals 60 degrees now if you're still messing up go back and what does it again make sure when you move 10 over your hitting the inverse Tan in typing in the number okay now let's look at some triangles and let's actually find this degree now they give please shut and they want us to find this angle measurements so if this right triangle is 16 and 27 you can use Pythagorean theorem to find that side what does this angle have to be for this Civic trunk so we have from the single this is the opposite  and for that angle this is the adjacent side  and of course this is my hypotenuse but they give me opposite and adjacent which one of our trig functions is off Jason it's tangent so we know that we're going to use tan and let's just say this angle is a tan of angle a equals opposite 16 religious 27 so now I need to get a bite self so move tangent over here so I can go equals the inverse tan of 16 over 27 type that in your calculator and you should get 30.65 it says nearest degree so that's going to round up 31 degrees so this angle is 31 degrees you can start from 90 and find the other one  which would happened to be 59 degrees eyeless look at number 6 now if you still need to label sides do it from this angle this is the adjacent cuz it's touching this is the opposite is across from the 90 it's the hypotenuse so I have hypotenuse and I have opposite which one is that that is sign so I know the sign and name it angle a again is opposite 46 or hypotenuse 48 so I know I'm going to do the inverse sine of 46 48 to get my a type that in your calculator and you should get 73.4 so that's going to round down to 70 3 degrees all right let's look at number 7 okay what do we have here we have the Jason side and we have the opposite side now what's opposite over adjacent that's tangent so I know going to do the tangent of this angle hey opposite is 15 at Jason's 21 so I know that I'm going to take the inverse tan because I remove the tan over here 18 / 21 type that in your calculator  and you are going to get 35.53 so that rounds up to 36 degrees now I want you to pause the video and I want you to do the next 3 on your own and then come back and see if you got them correct okay so for number 8 you had to sign and you should have gotten 17 degrees and then for number 9 and 10 you noticed they gave you all three sides you could have used any of them sine cosine or tangent so I set up which one and then you would have done the inverse that you should have gotten 23 degrees no matter which one you used and number 10 same thing I set up all of them you should have gotten 60 add degrees whichever one you got now if you need to pause on this and study this a little more do that let's go on to the next page okay so now we're going to bring this all together where you going to solve right triangles completely solve them that means to find all unknown side lengths and angle measures you can solve a right triangle when you know either the falling to side but you do Factory in theorem to start maybe or one side length in the measure of one acute angle that would be trig to start we're going to combine everything that we've learned I'm fine every side and every angle of a right triangle no-sew methods for solving right triangles the first one we learn was Pythagorean theorem which was a squared plus b squared equals c squared + C had to be the hypotenuse you should know that very well by now the second one was geometric mean now I'm going to pause and draw out the triangle for just a second now the big thing here to remember was this big triangle had to be a right triangle and the the dude had to be drawn to the hypotenuse and then that broke it up and we got the altitude which split the hypotenuse into segments one and segment shoot now remember this was the entire hypotenuse of the big triangle this was the long leg and this was the short leg and we had two formulas involve the altitude the altitude squared equals Segment 1 x segment 2 because geometric me is x squared equals 8 tons B now then we had one for the leg it could be the long leg or the short leg whichever one you chose the leg time's the segment that was adjacent to the leg times the whole hypotenuse now one key thing is remember none of these are squared only this side is squared so the altitude the altitude squared this distance squared equals Segment 1 ton segment 2 and then the leg if you use the short leg the segment of Jason was Segment 1 x the whole hypotenuse or if you use the long leg the segment adjacent was segment 2 times the whole hypotenuse so this squared equal this portion times this portion then we had to special right triangles which were 45-45-90 and 30-60-90 know remember 45-45-90 it's softly so the legs are the exact same amount whatever one is the other it and then hypotenuse you took that leg amount whatever it was and * this reduce to  and then for the 30-60-90 everything was based on the short leg opposite of the 30 that was RX then the hype can use was double the short leg so you just multiply the short like by 2 and the long leg was too short leg whatever that amount was times the square root of and then we had trig so in our trig we had our sign which was the opposite leg over the hypotenuse cosine which was the adjacent leg over the hypotenuse and tangent which was the opposite leg over the adjacent leg and then we had inverse trig which was the exact same thing you just use the inverse of the trig to do it sign was up silver hypotenuse cosine was adjacent over hypotenuse and tangent was opposite over adjacent so we pull all of that in together to solve right triangle that means every side every angle so Nexus 7 through 10 solve the right triangles round decimal answers to the nearest tenth YouMail is in radical form if you have radical form already going in your triangle but do not ever mix radical and decimal in the same answer so if we look at our first one right here number seven we are given two shots if we already have two sides it's easy to find the third one just use Instagram theorem now we have no ankles so we cannot use trig yet you could find one of the angles first do you have two sides and then you can use trig to find the third side but it is completely up to you I am going to use the Pythagorean theorem so this is my hypotenuse that I need to find what here so that's my seat so to find AC I would do 6 squared + 12 squared equals x squared now you should calculator work it out 6 squared + 12 squared is 180 then you have to take the square root of each side and you get that equals 13.4 so this is 13.4 now we need to find the angle let's say let's find angle a first I am going to use my 12in my 6 cuz they're whole numbers you can use sine cosine whatever you want but if I use these that's opposite in adjacent so that is going to be tangent so if I do the tan of angle a opposite of 12 adjacent next move the tangent over here so angle a equals the inverse tan of 12/6 type that in your calculator measure of angle is 63.4 degrees 63.4 degrees to find angle C what would you do new trick I would just do 90-60 3.4 and you get  26.6 degrees so this is 20 6.6 degrees so I have this single this single I found every angle and every side now I want you to try to do number 8 on your own suppose the video Work It what didn't come back and check it so for Ed you should have gotten 72 and 4 angle D you should have gotten 16.3 degrees and then see 73.7 degrees now I use Pythagorean theorem to find a d I use sign for D and then I just tried to 494 see you could have done this differently is long as you're getting the same answers you're fine  okay let's look at number nine this time they give us an angle measure and one side so we can't use Pythagorean start we can use we can find angle in first that would be easy we can use trig and any of those whites so let's look at it let's fun angle home first so we're just simply going to do 90 - 52 and we're going to get 38° okay so now we can use either angle trig to find that now we do know that this is our hypotenuse we can find l m or M in first let's say fun m in first this side right here  so if only use angle 52 this is the opposite and that's the hypotenuse so I'm is opposite here so Define m in I'm going to do the sine of 52 degrees equals opposite X  over hypotenuse 3 now remember when I saw here I put this over one and I cross multiply so my ex equals to 3 times the sine of 52 degrees top that in your calculator and you should get 2.3 which would be 2.4 now remember were using plane sign here not in Versa don't accidentally hit the inverse sine right now we're using plain sight so we know that this is 2.4 so we found this side this angle or missing now is LM so if I use that same angle this is my Jason I mean is trig again you could use Pythagorean theorem but I'm going to choose to do trick so that would be cosine because I'm using my Jason a 52 grease equals the adjacent will make it a over the hypotenuse 3 now again put that over one cross multiply a cools three times the cosine of 52 degrees type it in your calculator and you should get 1.8 so this is one point so now I have every side and every angle again I want you to pause the video and I want you to try and do number 10 on your own and then come back and check it okay so you should have gotten that ZX is 32.5 zy is 37.2 an angle wise 61 now I chose 10 ZX and then I did Pythagorean theorem once I got that second side so you can look at this study it pause it okay last problem  a boat pulled in by a winch on a dock 12 ft above the deck of the boat so here's our 12 ft the doc is 12 ft and here's our winch down to the boat when the winch is fully extended 25ft what is the angle of elevation from the boat to the winch now elevation is from the bottom up remember to question is from the top down so my elevation would be from this angle up to here so we're trying to find this angle right here okay now from here I have my opposite side and I have my hypotenuse cuz this is not here would be my adjacent which I don't have so what is opposite in partners that is sign so we know the sign of this angle X degrees equals opposite 12 over the hypotenuse 25 now we need to get the what's UPS move the sign over here so we'll have the inverse sine of 12 over 25  Annex degrees would equal to that in your calculator remember in versus second sign you should get 28.7 degrees so this angle of elevation right here is 28.7 degrees now since we're having the STAAR test this weekend you'll either be there on a Tuesday or Thursday but not both but everybody will be there one day this is the lesson for the day you're there and your homework will be due Monday we will have a quiz over it so I will see you then

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9.6 Notes AP

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